Advertisements
Advertisements
प्रश्न
The pth, qth and rth terms of an A.P. are a, b, c respectively. Show that (q – r )a + (r – p )b + (p – q )c = 0
Advertisements
उत्तर
Let t and d be the first term and the common difference of the A.P. respectively.
The nth term of an A.P. is given by, an = t + (n – 1) d
Therefore,
ap = t + (p – 1) d = a … (1)
aq = t + (q – 1)d = b … (2)
ar = t + (r – 1) d = c … (3)
Subtracting equation (2) from (1), we obtain
(p – 1 – q + 1) d = a – b
⇒ (p – q) d = a – b
संबंधित प्रश्न
In an A.P., if pth term is 1/q and qth term is 1/p, prove that the sum of first pq terms is 1/2 (pq + 1) where `p != q`
If the sum of first p terms of an A.P. is equal to the sum of the first q terms, then find the sum of the first (p + q) terms.
If the sum of n terms of an A.P. is 3n2 + 5n and its mth term is 164, find the value of m.
A farmer buys a used tractor for Rs 12000. He pays Rs 6000 cash and agrees to pay the balance in annual installments of Rs 500 plus 12% interest on the unpaid amount. How much will be the tractor cost him?
The nth term of a sequence is given by an = 2n2 + n + 1. Show that it is not an A.P.
Which term of the sequence 12 + 8i, 11 + 6i, 10 + 4i, ... is purely imaginary?
The first term of an A.P. is 5, the common difference is 3 and the last term is 80; find the number of terms.
The 4th term of an A.P. is three times the first and the 7th term exceeds twice the third term by 1. Find the first term and the common difference.
If the sum of three numbers in A.P. is 24 and their product is 440, find the numbers.
Find the sum of the following arithmetic progression :
41, 36, 31, ... to 12 terms
Find the sum of the following serie:
101 + 99 + 97 + ... + 47
Find the sum of all those integers between 100 and 800 each of which on division by 16 leaves the remainder 7.
Solve:
25 + 22 + 19 + 16 + ... + x = 115
If Sn = n2 p and Sm = m2 p, m ≠ n, in an A.P., prove that Sp = p3.
Find the sum of all two digit numbers which when divided by 4, yields 1 as remainder.
If S1 be the sum of (2n + 1) terms of an A.P. and S2 be the sum of its odd terms, then prove that: S1 : S2 = (2n + 1) : (n + 1).
The sums of first n terms of two A.P.'s are in the ratio (7n + 2) : (n + 4). Find the ratio of their 5th terms.
If a2, b2, c2 are in A.P., prove that \[\frac{a}{b + c}, \frac{b}{c + a}, \frac{c}{a + b}\] are in A.P.
If a, b, c is in A.P., then show that:
b + c − a, c + a − b, a + b − c are in A.P.
If a, b, c is in A.P., then show that:
bc − a2, ca − b2, ab − c2 are in A.P.
A man arranges to pay off a debt of Rs 3600 by 40 annual instalments which form an arithmetic series. When 30 of the instalments are paid, he dies leaving one-third of the debt unpaid, find the value of the first instalment.
There are 25 trees at equal distances of 5 metres in a line with a well, the distance of the well from the nearest tree being 10 metres. A gardener waters all the trees separately starting from the well and he returns to the well after watering each tree to get water for the next. Find the total distance the gardener will cover in order to water all the trees.
A man is employed to count Rs 10710. He counts at the rate of Rs 180 per minute for half an hour. After this he counts at the rate of Rs 3 less every minute than the preceding minute. Find the time taken by him to count the entire amount.
In a potato race 20 potatoes are placed in a line at intervals of 4 meters with the first potato 24 metres from the starting point. A contestant is required to bring the potatoes back to the starting place one at a time. How far would he run in bringing back all the potatoes?
A man saved ₹66000 in 20 years. In each succeeding year after the first year he saved ₹200 more than what he saved in the previous year. How much did he save in the first year?
In a cricket team tournament 16 teams participated. A sum of ₹8000 is to be awarded among themselves as prize money. If the last place team is awarded ₹275 in prize money and the award increases by the same amount for successive finishing places, then how much amount will the first place team receive?
If 7th and 13th terms of an A.P. be 34 and 64 respectively, then its 18th term is
If four numbers in A.P. are such that their sum is 50 and the greatest number is 4 times the least, then the numbers are
Let Sn denote the sum of n terms of an A.P. whose first term is a. If the common difference d is given by d = Sn − k Sn − 1 + Sn − 2 , then k =
Mark the correct alternative in the following question:
\[\text { If in an A . P } . S_n = n^2 q \text { and } S_m = m^2 q, \text { where } S_r \text{ denotes the sum of r terms of the A . P . , then }S_q \text { equals }\]
If second, third and sixth terms of an A.P. are consecutive terms of a G.P., write the common ratio of the G.P.
If a, b, c are in A.P. and x, y, z are in G.P., then the value of xb − c yc − a za − b is
Find the sum of first 24 terms of the A.P. a1, a2, a3, ... if it is known that a1 + a5 + a10 + a15 + a20 + a24 = 225.
The product of three numbers in A.P. is 224, and the largest number is 7 times the smallest. Find the numbers
A man accepts a position with an initial salary of Rs 5200 per month. It is understood that he will receive an automatic increase of Rs 320 in the very next month and each month thereafter. Find his salary for the tenth month
If the sum of n terms of an A.P. is given by Sn = 3n + 2n2, then the common difference of the A.P. is ______.
Let Sn denote the sum of the first n terms of an A.P. If S2n = 3Sn then S3n: Sn is equal to ______.
If a1, a2, a3, .......... are an A.P. such that a1 + a5 + a10 + a15 + a20 + a24 = 225, then a1 + a2 + a3 + ...... + a23 + a24 is equal to ______.
If b2, a2, c2 are in A.P., then `1/(a + b), 1/(b + c), 1/(c + a)` will be in ______
The internal angles of a convex polygon are in A.P. The smallest angle is 120° and the common difference is 5°. The number to sides of the polygon is ______.
